EMF, Terminal Potential Difference and Internal Resistance of a Cell

The electromotive force (EMF) of a cell, denoted $\varepsilon$, is defined as the potential difference between its two terminals when NO current is being drawn from it (an open circuit) -- equivalently, it is the total amount of work the cell's internal chemical (or other) energy source does, per unit charge, in driving charge around a complete circuit, including through the cell itself.

A real cell, however, is not a perfect source: the chemicals and electrodes inside it always offer some opposition to the flow of current, called the cell's internal resistance, $r$. The moment current is actually drawn from the cell (i.e. the circuit is closed through some external resistance $R$), part of the cell's own EMF is used up in driving the current through this internal resistance, and only the remainder appears as the usable potential difference across the cell's own terminals -- the terminal potential difference, $V$.

For a cell of EMF $\varepsilon$ and internal resistance $r$, connected to an external resistance $R$, the current in the circuit is found by applying Ohm's law to the complete loop (external resistance plus internal resistance in series, driven by the EMF):

and the terminal potential difference, the potential difference actually measurable across the cell's own terminals (equivalently, across $R$, since they are directly connected), works out to

So the terminal potential difference is always LESS than the EMF whenever current flows (i.e. whenever $I>0$), by exactly the "lost volts" $Ir$ dropped across the cell's own internal resistance. Only in the special limiting cases of an open circuit ($I=0$, so $V=\varepsilon$ exactly) or an ideal cell with zero internal resistance ($r=0$) does the terminal potential difference equal the EMF.

If the cell is instead being CHARGED (current forced through it in the reverse direction by some external source of higher EMF), the sign of the $Ir$ term reverses, and the terminal potential difference EXCEEDS the EMF: $V = \varepsilon + Ir$ -- a useful check that shows the "lost volts" always opposes whichever direction the current is being pushed relative to the cell's own EMF. …